Jacobi Spectral Collocation Method For Riesz - Type Fractional Differential Equations

In the past several decades, spectral method as one of the important tools for scientific computing developed rapidly. In this thesis, we provide an effective new algorithm for the space or time-space Reisz-type fractional differential equation. We extend the unilateral RiemannLiouville（R-L） fractional pseudospectral differential matrix in [32] to the generally fractional pseudospectral differential matrix with R-L fractional derivative on both sides, so as to construct the first kind of Riesz-type fractional pseudospectral differential matrix. For the second kind of Riesz-type fractional derivative, we use the Jacobi polynomial （1-x²）^s/2p₂⁽（s/2,s/2）（x）n（x） to construct pseudospectral differential matrix. It is to use the singularity of the Jacobi weight （1-x²）^s/2 to match the singularity of the second type of Riesz-type differential operator. We construct interpolating basis function of the second kind of Riesz-type fractional differential operator by using the Birkhoff interpolation. It’s matrix is the exact inverse（in the inner nodes） of the pseudospectral differential matrix. Then, we get a stable discrete system. In this thesis we also provide abundant numerical results to show that our new numerical method is effective.